3.290 \(\int \frac{x^6 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=336 \[ -\frac{x \left (19 a^2 b e-25 a^3 f-13 a b^2 d+7 b^3 c\right )}{18 b^5 \left (a+b x^3\right )}+\frac{a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{54 a^{2/3} b^{16/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{27 a^{2/3} b^{16/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{2/3} b^{16/3}}+\frac{x \left (6 a^2 f-3 a b e+b^2 d\right )}{b^5}+\frac{x^4 (b e-3 a f)}{4 b^4}+\frac{f x^7}{7 b^3} \]

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x)/b^5 + ((b*e - 3*a*f)*x^4)/(4*b^4) + (f*x^7)/(7*b^3) + (a*(b^3*c - a*b^2*d + a^
2*b*e - a^3*f)*x)/(6*b^5*(a + b*x^3)^2) - ((7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(18*b^5*(a + b*x^
3)) - ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sq
rt[3]*a^(2/3)*b^(16/3)) + ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(2/3
)*b^(16/3)) - ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
(54*a^(2/3)*b^(16/3))

________________________________________________________________________________________

Rubi [A]  time = 0.508736, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1828, 1858, 1887, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (19 a^2 b e-25 a^3 f-13 a b^2 d+7 b^3 c\right )}{18 b^5 \left (a+b x^3\right )}+\frac{a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{54 a^{2/3} b^{16/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{27 a^{2/3} b^{16/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{2/3} b^{16/3}}+\frac{x \left (6 a^2 f-3 a b e+b^2 d\right )}{b^5}+\frac{x^4 (b e-3 a f)}{4 b^4}+\frac{f x^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b^2*d - 3*a*b*e + 6*a^2*f)*x)/b^5 + ((b*e - 3*a*f)*x^4)/(4*b^4) + (f*x^7)/(7*b^3) + (a*(b^3*c - a*b^2*d + a^
2*b*e - a^3*f)*x)/(6*b^5*(a + b*x^3)^2) - ((7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(18*b^5*(a + b*x^
3)) - ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sq
rt[3]*a^(2/3)*b^(16/3)) + ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(2/3
)*b^(16/3)) - ((2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/
(54*a^(2/3)*b^(16/3))

Rule 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol
ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +
1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[
a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q
- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^6 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\int \frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-6 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^6-6 a b^3 (b e-a f) x^9-6 a b^4 f x^{12}}{\left (a+b x^3\right )^2} \, dx}{6 a b^5}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac{\int \frac{2 a^2 b^4 \left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right )+18 a^2 b^5 \left (b^2 d-2 a b e+3 a^2 f\right ) x^3+18 a^2 b^6 (b e-2 a f) x^6+18 a^2 b^7 f x^9}{a+b x^3} \, dx}{18 a^2 b^9}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac{\int \left (18 a^2 b^4 \left (b^2 d-3 a b e+6 a^2 f\right )+18 a^2 b^5 (b e-3 a f) x^3+18 a^2 b^6 f x^6-\frac{2 \left (-2 a^2 b^7 c+14 a^3 b^6 d-35 a^4 b^5 e+65 a^5 b^4 f\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^9}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac{(b e-3 a f) x^4}{4 b^4}+\frac{f x^7}{7 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{9 b^5}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac{(b e-3 a f) x^4}{4 b^4}+\frac{f x^7}{7 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{2/3} b^5}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{2/3} b^5}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac{(b e-3 a f) x^4}{4 b^4}+\frac{f x^7}{7 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{2/3} b^{16/3}}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 \sqrt [3]{a} b^5}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac{(b e-3 a f) x^4}{4 b^4}+\frac{f x^7}{7 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{2/3} b^{16/3}}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{2/3} b^{16/3}}\\ &=\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac{(b e-3 a f) x^4}{4 b^4}+\frac{f x^7}{7 b^3}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^5 \left (a+b x^3\right )^2}-\frac{\left (7 b^3 c-13 a b^2 d+19 a^2 b e-25 a^3 f\right ) x}{18 b^5 \left (a+b x^3\right )}-\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{16/3}}+\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{16/3}}-\frac{\left (2 b^3 c-14 a b^2 d+35 a^2 b e-65 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{2/3} b^{16/3}}\\ \end{align*}

Mathematica [A]  time = 0.313184, size = 323, normalized size = 0.96 \[ \frac{-\frac{42 \sqrt [3]{b} x \left (19 a^2 b e-25 a^3 f-13 a b^2 d+7 b^3 c\right )}{a+b x^3}+\frac{126 a \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-35 a^2 b e+65 a^3 f+14 a b^2 d-2 b^3 c\right )}{a^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-65 a^3 f-14 a b^2 d+2 b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-35 a^2 b e+65 a^3 f+14 a b^2 d-2 b^3 c\right )}{a^{2/3}}+756 \sqrt [3]{b} x \left (6 a^2 f-3 a b e+b^2 d\right )+189 b^{4/3} x^4 (b e-3 a f)+108 b^{7/3} f x^7}{756 b^{16/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(756*b^(1/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x + 189*b^(4/3)*(b*e - 3*a*f)*x^4 + 108*b^(7/3)*f*x^7 + (126*a*b^(1/3
)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3)^2 - (42*b^(1/3)*(7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a
^3*f)*x)/(a + b*x^3) + (28*Sqrt[3]*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a
^(1/3))/Sqrt[3]])/a^(2/3) + (28*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/
3) + (14*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/
3))/(756*b^(16/3))

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 596, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

-14/27/b^4*a*d/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+7/27/b^4*a*d/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/
3))+2/27/b^3*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+13/18/b^2/(b*x^3+a)^2*x^4*a*d-1
/27/b^3*c/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-3/4/b^4*x^4*a*f+6/b^5*a^2*f*x-3/b^4*a*e*x-7/18/b
/(b*x^3+a)^2*x^4*c+2/27/b^3*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+1/7*f*x^7/b^3+1/4/b^3*x^4*e+1/b^3*d*x+35/27/b^
5*a^2*e/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-35/54/b^5*a^2*e/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-
2/9/b^2/(b*x^3+a)^2*a*c*x+25/18/b^4/(b*x^3+a)^2*x^4*a^3*f-19/18/b^3/(b*x^3+a)^2*x^4*a^2*e-65/27/b^6*a^3*f/(1/b
*a)^(2/3)*ln(x+(1/b*a)^(1/3))+65/54/b^6*a^3*f/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+5/9/b^3/(b*x
^3+a)^2*a^2*d*x+11/9/b^5/(b*x^3+a)^2*a^4*f*x-8/9/b^4/(b*x^3+a)^2*a^3*e*x-65/27/b^6*a^3*f/(1/b*a)^(2/3)*3^(1/2)
*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+35/27/b^5*a^2*e/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)
^(1/3)*x-1))-14/27/b^4*a*d/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.46143, size = 2986, normalized size = 8.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/756*(108*a^2*b^5*f*x^13 + 27*(7*a^2*b^5*e - 13*a^3*b^4*f)*x^10 + 54*(14*a^2*b^5*d - 35*a^3*b^4*e + 65*a^4*b
^3*f)*x^7 - 147*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^4 - 42*sqrt(1/3)*(2*a^3*b^4*c - 1
4*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f + (2*a*b^6*c - 14*a^2*b^5*d + 35*a^3*b^4*e - 65*a^4*b^3*f)*x^6 + 2*(2*
a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^3)*sqrt((-a^2*b)^(1/3)/b)*log((2*a*b*x^3 + 3*(-a^2*b
)^(1/3)*a*x - a^2 - 3*sqrt(1/3)*(2*a*b*x^2 + (-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt((-a^2*b)^(1/3)/b))/(b*x
^3 + a)) - 14*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*
b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-
a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*
c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*(-a^2
*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) - 84*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f)*x)/(a^2*b^
8*x^6 + 2*a^3*b^7*x^3 + a^4*b^6), 1/756*(108*a^2*b^5*f*x^13 + 27*(7*a^2*b^5*e - 13*a^3*b^4*f)*x^10 + 54*(14*a^
2*b^5*d - 35*a^3*b^4*e + 65*a^4*b^3*f)*x^7 - 147*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^
4 + 84*sqrt(1/3)*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f + (2*a*b^6*c - 14*a^2*b^5*d + 35*a^3*
b^4*e - 65*a^4*b^3*f)*x^6 + 2*(2*a^2*b^5*c - 14*a^3*b^4*d + 35*a^4*b^3*e - 65*a^5*b^2*f)*x^3)*sqrt(-(-a^2*b)^(
1/3)/b)*arctan(sqrt(1/3)*(2*(-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt(-(-a^2*b)^(1/3)/b)/a^2) - 14*((2*b^5*c -
14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4*b*e - 65*a^5*f + 2*(2*a*b
^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)
^(1/3)*a) + 28*((2*b^5*c - 14*a*b^4*d + 35*a^2*b^3*e - 65*a^3*b^2*f)*x^6 + 2*a^2*b^3*c - 14*a^3*b^2*d + 35*a^4
*b*e - 65*a^5*f + 2*(2*a*b^4*c - 14*a^2*b^3*d + 35*a^3*b^2*e - 65*a^4*b*f)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a
^2*b)^(2/3)) - 84*(2*a^3*b^4*c - 14*a^4*b^3*d + 35*a^5*b^2*e - 65*a^6*b*f)*x)/(a^2*b^8*x^6 + 2*a^3*b^7*x^3 + a
^4*b^6)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.10782, size = 539, normalized size = 1.6 \begin{align*} -\frac{{\left (2 \, b^{3} c - 14 \, a b^{2} d - 65 \, a^{3} f + 35 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{5}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{6}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 65 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a b^{6}} - \frac{7 \, b^{4} c x^{4} - 13 \, a b^{3} d x^{4} - 25 \, a^{3} b f x^{4} + 19 \, a^{2} b^{2} x^{4} e + 4 \, a b^{3} c x - 10 \, a^{2} b^{2} d x - 22 \, a^{4} f x + 16 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{5}} + \frac{4 \, b^{18} f x^{7} - 21 \, a b^{17} f x^{4} + 7 \, b^{18} x^{4} e + 28 \, b^{18} d x + 168 \, a^{2} b^{16} f x - 84 \, a b^{17} x e}{28 \, b^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*(2*b^3*c - 14*a*b^2*d - 65*a^3*f + 35*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5) + 1/27*sq
rt(3)*(2*(-a*b^2)^(1/3)*b^3*c - 14*(-a*b^2)^(1/3)*a*b^2*d - 65*(-a*b^2)^(1/3)*a^3*f + 35*(-a*b^2)^(1/3)*a^2*b*
e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^6) + 1/54*(2*(-a*b^2)^(1/3)*b^3*c - 14*(-a*b^2)^
(1/3)*a*b^2*d - 65*(-a*b^2)^(1/3)*a^3*f + 35*(-a*b^2)^(1/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/
(a*b^6) - 1/18*(7*b^4*c*x^4 - 13*a*b^3*d*x^4 - 25*a^3*b*f*x^4 + 19*a^2*b^2*x^4*e + 4*a*b^3*c*x - 10*a^2*b^2*d*
x - 22*a^4*f*x + 16*a^3*b*x*e)/((b*x^3 + a)^2*b^5) + 1/28*(4*b^18*f*x^7 - 21*a*b^17*f*x^4 + 7*b^18*x^4*e + 28*
b^18*d*x + 168*a^2*b^16*f*x - 84*a*b^17*x*e)/b^21